In this problem we will calculate the electric field of a li

In this problem, we will calculate the electric field of a line charge. The line charge is aligned along the x-axis starting at the origin and having a length L. The line has a linear charge density ?. We want to find the electric field at a point P on the x-axis. The point P is located at the point (d, 0) where d > L.

(a) List all of the given parameters.

(b) What is the dimensionality of a line of charge? What is the electric field of a small charge dq which is one dimension less than our line?

(c) What is the charge dq in terms of the given parameters?

(d) What is the distance between an arbitrary point on our line charge to the point P in terms of given parameters?

(f) What is the expression for the electric field of the line in integral form? This means plug in your answers to parts (c), (d), and (e) into your answer for part (b) but don’t do the integral yet.

(g) Rewrite this expression with all the constants pulled out of the integral. Include the bounds of the integral if you have not done so already.

(h) What is the electric field of this line charge?

Solution

a) Given:
Length = L
Charge density = C
Location of point = (d,0)
b) One Dimension
dE = kdQ/r^2
where r is the distance from the dQ charge

c) dQ = CdL
d) r = d - x (where x is the location of the arbitrary dQ from the origin)
e) r^ = r / | r | = 1
f) E = integral(kCdx/(d-x)^2) from x = 0 to x = L
g) E = kC*integral(1dx/(d-x)^2) from x = 0 to x = L
h) E = kC*L/[d*(d-L)]